{
  "id": "2209.06401",
  "version": "v1",
  "published": "2022-09-14T03:50:46.000Z",
  "updated": "2022-09-14T03:50:46.000Z",
  "title": "Ryser's Theorem for Symmetric $ρ$-latin Squares",
  "authors": [
    "Amin Bahmanian"
  ],
  "comment": "16 pages. arXiv admin note: text overlap with arXiv:2201.04793",
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "Let $L$ be an $n\\times n$ array whose top left $r\\times r$ subarray is filled with $k$ different symbols, each occurring at most once in each row and at most once in each column. We establish necessary and sufficient conditions that ensure the remaining cells of $L$ can be filled such that each symbol occurs at most once in each row and at most once in each column, $L$ is symmetric with respect to the main diagonal, and each symbol occurs a prescribed number of times in $L$. The case where the prescribed number of times each symbol occurs is $n$ was solved by Cruse (J. Combin. Theory Ser. A 16 (1974), 18-22), and the case where the top left subarray is $r\\times n$ and the symmetry is not required, was settled by Goldwasser et al. (J. Combin. Theory Ser. A 130 (2015), 26-41). Our result allows the entries of the main diagonal to be specified as well, which leads to an extension of the Andersen-Hoffman's Theorem (Annals of Disc. Math. 15 (1982) 9-26, European J. Combin. 4 (1983) 33-35).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-14T03:50:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B15",
      "05C70",
      "05C15"
    ],
    "keywords": [
      "latin squares",
      "rysers theorem",
      "symbol occurs",
      "theory ser",
      "main diagonal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}